Optimal allocation with ex post verification and limited penalties

Several agents with privately known social values compete for a prize. The prize is allocated based on the claims of the agents, and the winner is subject to a limited penalty if he makes a false claim. If the number of agents is large, the optimal mechanism places all agents above a threshold onto a shortlist along with a fraction of agents below the threshold, and then allocates the prize to a random agent on the shortlist. When the number of agents is small, the optimal mechanism allocates the prize to the agent who makes the highest claim, but restricts the range of claims above and below.