Mechanism design with costly verification and limited punishments

A principal allocates an object (e.g., a job or grant) among several agents, each of whom wants the object and privately knows the value to the principal of assigning it to him. The object is allocated based on the agents' reports. The principal can choose to inspect an agent's report at a cost and impose a limited punishment on the one who receives the object. An optimal mechanism specifies two thresholds v^l≤v^u. If every agent reports a value below v^l, the object is assigned to a random agent, and no one is inspected. If any agents report a value above v^l but all reports are below v^u, the agent with the highest reported value receives the object and is inspected with some probability. If any agents report values above v^u, one of them is randomly selected to receive the object and is inspected with certainty. An agent is punished if and only if his report is found to be false. When the number of agents is small, v^u is equal to the upper bound of the support of the value distribution. When the number of agents is large, v^l=v^u.