Auctions with a minimum requirement of bids

We show that in a procurement auction with independent and private costs of production and a positive cost of preparing a bid, the requirement of a minimum number of offers for the good to be bought always yields a unique (perfect) Bayesian equilibrium where no firm enters a bid, whatever its cost of production, the number of potential bidders and the size of the bidding cost. To avoid the no-bid result, the buyer can commit to subsidise the losing bidders in certain circumstances. Alternatively, it can use a stochastic auction, where the provider of the good is not always the firm that bids the lowest price.