E ffi cient , strategy-proof and almost budget-balanced assignment

Call a Vickrey-Clarke-Groves (VCG) mechanism to assign p identical 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 negative valuations. The optimal (smallest) efficiency loss L(n, p) satisfies L(n, p) ≤ L(n, { 2 }) ≤ 4 3 √ n for all n. If p n is strictly smaller or strictly larger than 1 2 , the convergence of L(n, p) to zero is exponential in n. The optimal mechanism achieving L(n, p) is individually rational (participation is voluntary) if p = 1, but not if p ≥ 2. Among all feasible and voluntary mechanisms, the optimal efficiency loss L∗(n, p) is not significantly larger than L(n, p) if p n ≤ 2. But it does not converge to zero if p n > 2. 1 The problem and the punch line A group of n agents must assign p identical objects, where p < n. These objects are desirable so we talk of “goods” and we have a simple rationing problem: each agent claims a unit but all claims can’t be met. For instance the n friends share p tickets to a popular show; the city assigns a few alcohol licenses to bar owners; a school principal distributes scarce computers to classrooms, or a manager distributes fancy machines in short supply between several divisions. Cash transfers are used to compensate the losers (who get no object) from the winner’s pocket, and to align incentives and efficiency. Preferences are quasi-linear in money, described by a non negative valuation (willingness to pay) for an object. A Vickrey-Clarke-Groves (VCG) mechanism performs monetary transfers inducing truthful revelation of individual valuations and implements an efficient assignment, i.e., assigns the objects to the p ∗Thanks to Sandeep Baliga, Geoffroy de Clippel, Olivier Gossner, Jason Hartline, and Jay Sethuraman for their comments.