Optimal Strategy-Proof and Budget Balanced Mechanisms to Assign Multiple Objects

Abstract Strategy-proof, budget balanced, and envy-free linear mechanisms assign p identical objects to n agents. The efficiency loss is the largest ratio of surplus loss to efficient surplus, over all profiles of non-negative valuations. The smallest efficiency loss \frac{n-p}{n^{2}-n} is uniquely achieved by the following simple allocation rule: assign one object to each of the p-1 agents with the highest valuations, a large probability to the agent with the pth highest valuation, and the remaining probability to the agent with the (p 1)th highest valuation.When “envy freeness” is replaced by the weaker condition “voluntary participation”, optimal mechanisms differ only when p is less than a threshold, which depends on n and is less than \frac{n}{2}. For p fixed, the smallest efficiency loss converges to zero exponentially in n