(Sub-)optimal entry fees

We extend Myerson's (1981) model by allowing for uncertainty about the number of bidders. In such extension the Revenue Equivalence Theorem still holds and the optimal allocation rule remains the same. Hence, the optimal auction can be implemented with an appropriate reserve price. Nonetheless, we show that entry fees are sub-optimal. The reasons are heterogeneity in bidders' beliefs about the number of bidders, and auctioneer's uncertainty about the optimum entry fee, if any. Our result implies a reversal of the revenue ranking by Milgrom and Weber (1982) which is consistent with many real life situations: auction houses, internet auctions,..