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 satisfies . If is bounded away from , converges to zero exponentially in...

Agents with single-peaked preferences share a resource coming from different suppliers; each agent is connected to only a subset of suppliers. Examples include workload balancing, sharing earmarked funds, and rationing utilities after a storm.