Efficiency and Budget Balance in General Quasi-linear Domains

We study efficiency and budget balance for designing mechanisms in general quasi-linear domains. Green and Laffont (1979) proved that one cannot generically achieve both. We consider strategyproof budget-balanced mechanisms that are approximately efficient. For deterministic mechanisms, we show that a strategyproof and budget-balanced mechanism must have a sink agent whose valuation function is ignored in selecting an alternative, and she is compensated with the payments made by the other agents. We assume the valuations of the agents come from a bounded open interval. This result strengthens Green and Laffont’s impossibility result by showing that even in a restricted domain of valuations, there does not exist a mechanism that is strategyproof, budget balanced, and takes every agent’s valuation into consideration—a corollary of which is that it cannot be efficient. Using this result, we find a tight lower bound on the inefficiencies of strategyproof, budget-balanced mechanisms in this domain. The bound shows that the inefficiency asymptotically disappears when the number of agents is large—a result close in spirit to Green and Laffont (1979, Theorem 9.4). However, our results provide worst-case bounds and the best possible rate of convergence. Next, we consider minimizing any convex combination of inefficiency and budget imbalance. We show that if the valuations are unrestricted, no deterministic mechanism can do asymptotically better than minimizing inefficiency alone. Finally, we investigate randomized mechanisms and provide improved lower bounds on expected inefficiency. We give a tight lower bound for an interesting class of strategyproof, budgetbalanced, randomized mechanisms. We also use an optimization-based approach—in the spirit of automated mechanism design—to provide a lower bound on the minimum achievable inefficiency of any randomized mechanism. Experiments with real data from two applications show that the inefficiency for a simple randomized mechanism is 5–100 times smaller than the worst case. This relative difference increases with the number of agents. JEL Codes: D82, D71, D74