Revenue in contests with many participants

We show that in a contest with a single prize, the expected effort made by the kkth highest valuation participant bounds the sum of the expected efforts made by all of the participants with valuations less than the kkth highest valuations. We also show that in the limit case of a contest with mm prizes, the expected effort made by the kkth highest valuation participant when the bidders are risk-neutral is greater than the expected effort in the risk-averse case.